Résumé de l'exposé
Recent years have seen much progress in boundary value problems for elliptic operators in non-smooth settings with $L^p$ boundary data. In particular, we now have a good understanding of solvability of the $L^p$ Dirichlet problem and many of its characterizations. There have also been big breakthroughs recently on the Regularity problem, which is a Dirichlet problem with $W^{1,p}$ boundary data. However, little progress has been made on the Neumann problem since the works of Kenig and Pipher in the mid 90s. In a joint work with Joseph Feneuil, we introduce the $L^p$ Poisson-Neumann problem and its variants, with the hope that it can serve as a stepping stone to eventually solving the Neumann problem. In the talk, I will discuss some characterizations of the Poisson-Neumann problem and its weaker variants, their connections to the Neumann problem, and will show that an extrapolation result on the Neumann problem obtained by Kenig and Pipher can be improved with the help of the Poisson-Neumann problem.